On the combinatorics of sparsification

TL;DR

This paper analyzes the combinatorial aspects of sparsification in RNA folding algorithms, showing significant theoretical and experimental reductions in computational complexity for various energy models.

Contribution

It introduces a combinatorial framework to quantify sparsification effects in RNA structure folding, including expected candidate set sizes and complexity reductions.

Findings

Sparsification reduces RNA folding time complexity by up to 98%.

The framework predicts candidate set sizes using probabilistic models.

Experimental results confirm theoretical complexity reductions.

Abstract

Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, $\Lambda^*$, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of $\Lambda^*$-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of $\Lambda^*$-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the "full" loop-energy model there is a reduction of 98% (experiment).